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CHAPTER 50 DEFLECTION Joseph E. Shigley Professor Emeritus The University of Michigan Ann Arbor, Michigan Charles R. Mischke, Ph.D., RE. Professor Emeritus of Mechanical Engineering Iowa State University Ames, Iowa 50.1 STIFFNESS OR SPRING RATE / 50.2 50.2 DEFLECTION DUE TO BENDING / 50.3 50.3 PROPERTIES OF BEAMS / 50.3 50.4 COMPUTER ANALYSIS / 50.3 50.5 ANALYSIS OF FRAMES /50.15 GLOSSARY OF SYMBOLS a Dimension A Area b Dimension C Constant Dy d Diameter E Young's modulus F Force G Shear modulus / Second moment of area / Second polar moment of area k Spring rate K Constant € Length M Moment M(J) Moment relation, (MIEf) 1 N Number q Unit load Q Fictitious force R Support reaction T Torque U Strain energy V Shear force w Unit weight W Total weight jc Coordinate y Coordinate 8 Deflection 0 Slope, torsional deflection <|> An integral \j/ An integral 50.7 STIFFNESSORSPRINGRATE The spring rate (also called stiffness or scale) of a body or ensemble of bodies is defined as the partial derivative of force (torque) with respect to colinear displace- ment (rotation). For a helical tension or compression spring, Z7 ^Gy ^ j 3F d*G , <ni , F= 8DW thUS * = > = 8^W (5ai) where D = mean coil diameter d = wire diameter N = number of active turns In a round bar subject to torsion, T = m thus * = |I = ^ (50.2) and the tensile force in an elongating bar of any cross section is ,, AES ,, , 3F AE /cn -v F =~T thus k= K = — (503) If fc is constant, as in these cases, then displacement is said to be linear with respect to force (torque). For contacting bodies with all four radii of curvature finite, the approach of the bodies is proportional to load to the two-thirds power, making the spring rate proportional to load to the one-third power. In hydrodynamic film bear- ings, the partial derivative would be evaluated numerically by dividing a small change in load by the displacement in the direction of the load. 50.2 DEFLECTIONDUETOBENDING ___^ The relations involved in the bending of beams are well known and are given here for reference purposes as follows: ir£ <»*> %-% £-0 < 5ft6 > e = f «50.7, f-M (50.8) These relations are illustrated by the beam of Fig. 50.1. Note that the x axis is posi- tive to the right and the y axis is positive upward. All quantities—loading, shear force, support reactions, moment, slope, and deflection—have the same sense as y; they are positive if upward, negative if downward. 50.3 PROPERTIESOFBEAMS Table 50.1 lists a number of useful properties of beams having a variety of loadings. These must all have the same cross section throughout the length, and a linear rela- tion must exist between the force and the deflection. Beams having other loadings can be solved using two or more sets of these relations and the principle of super- position. In using Table 50.1, remember that the deflection at the center of a beam with off- center loads is usually within 2.5 percent of the maximum value. 50.4 COMPUTERANALYSIS In this section we will develop a computer method using numerical analysis to deter- mine the slope and deflection of any simply supported beam having a variety of con- centrated loads, including point couples, with any number of step changes in cross section. The method is particularly applicable to stepped shafts where the transverse bending deflections and neutral-axis slopes are desired at specified points. The method uses numerical analysis to integrate Eq. (50.6) twice in a marching method. The first integration uses the trapezoidal rule; the second uses Simpson's rule (see Sec. 4.6). The procedure gives exact results. Let us define the two successive integrals as $=\ X ^f J dx v= fW (50.9) J o &i J o FIGURE 50.1 (a) Loading diagram showing beam supported at A and B with uniform load w having units of force per unit length, Ri = R 2 = w€/2; (b) shear-force diagram showing end con- ditions; (c) moment diagram; (d) slope diagram; (e) deflection diagram. But from Eq. (50.7), the slope is e ^.f«fc + Cl ax J 0 EI = <» +C 1 (a) TABLE 50.1 Properties of Beams 1. Cantilever—intermediate load 2. Cantilever—intermediate couple TABLE 50.1 Properties of Beams (Continued) 3. Cantilever—distributed load 4. Cantilever—partial distributed load 5. Cantilever—partial distributed load 6. Simple support—intermediate load TABLE 50.1 Properties of Beams (Continued) 7. Simple support—intermediate couple 8. Simple support—end moments 9. Simple support—overhung load 10. Simple support-— uniform loading TABLE 50.1 Properties of Beams (Continued) 11. Simple support—partial uniform loading 12. Simple support—uniform loading, overhung